On the homotopy groups of separable metric spaces

The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. First, we state a nerve theorem. It gives sufficient conditions under which there is a weak n-equivalence between the nerve of a good cover and its underlying space. Then we conclude that for any (n−1)-connected, locally (n−1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, πn(X), is finitely presented. This result allows us to provide a new proof for a generalization of Shelah?s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topological space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite which introduce here. Finally, we prove that each 2-loop is homotopic to a reduced 2-loop.     

Geometric versus homotopy theoretic equivariant bordism

By results of Löffler and Comezaña, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G=S¹××S¹, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of Sinha [22]. This is used in order to determine the image of the Pontrjagin-Thom map for toralG.     

Vertex-minimal simplicial immersions of the Klein bottle in three space

Although the Klein bottle cannot be embedded inR ³, it can be immersed there, and in more than one way. Smooth examples of these immersions have been studied extensively, but little is known about their simplicial versions. The vertices of a triangulation play a crucial role in understanding immersions, so it is reasonable to ask: How few vertices are required to immerse the Klein bottle inR ³? Several examples that use only nine vertices are given in Section 3, and since any triangulation of the Klein bottle must have at least eight vertices, the question becomes: Can the Klein bottle be immersed inR ³ using only eight vertices? In this paper, we show that, in fact, eight isnot enough, nine are required. The proof consists of three parts: first exhibiting examples of 9-vertex immersions; second determining all possible 8-vertex triangulations ofK ²; and third showing that none of these can be immersed inR ³.     

Self homotopy groups with large nilpotency classes

We demonstrate that for any n>0 there exists a compact connected Lie group G such that the self homotopy group [G,G] has the nilpotency class greater than n, where [G,G] is a nilpotent group for a compact connected Lie group G.     

Simple homotopy types and finite spaces

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X↘Y of finite spaces induces a simplicial collapse K(X)↘K(Y) of their associated simplicial complexes. Moreover, a simplicial collapse K↘L induces a collapse X(K)↘X(L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.     

On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A (G,n)-complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n−1. In case G is an n-dimensional group there is a unique (up to homotopy) (G,n)-complex on the minimal Euler-characteristic level χmin(G,n). For every n we give examples of n-dimensional groups G for which there exist homotopically distinct (G,n)-complexes on the level χmin(G,n)+1. In the case where n=2 these examples are algebraic.     

A sufficient condition for a set of calibrated surfaces to be area-minimizing under diffeomorphisms

We extend Choe’s idea in [1] to nonpolyhedral calibrated surfaces and give some examples of polyhedral sets over right prisms and nonpolyhedral calibrated surfaces. Received: 4 October 2004     

A non-commutative formula for the colored Jones function

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern–Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse of a generalized determinant. Our formula also reveals a direct relation between the Alexander polynomial and the colored Jones function of a knot and immediately implies the extensively studied Melvin–Morton–Rozansky conjecture, first proved by Bar–Natan and the first author about 10 years ago. Our results complement recent work of Huynh and Le, who also give a non-commutative formulae for the colored Jones function of a knot, starting from a non-commutative formula for the R matrix of the quantum group ; see Huynh and Le (in math.GT/0503296).     

A graph-theoretical representation of PL-manifolds — A survey on crystallizations

This is a survey of the techniques and results developed by M. Pezzana and his group, which includes, besides the authors, A. Cavicchioli, P. Bandieri and A Donati.The original concept is that of contracted triangulation, which was introduced with the main goal of finding a minimal atlas for topological manifolds ([P₁ 1968], [P₂ 1974], [P₃ 1974], [FG₂ 1979]). Only later did the possibility of deducing a graph-theoretical tool — the crystallization — for representing P.L. manifolds occur as a major aspect of the theory ([P₄ 1975], [F₁ 1976]). This leads to an application of graph theory to P.L. topology, which seems not to have been explored before. Recently, other authors outside Italy have independently become interested in this subject.For the sake of conciseness, definitions and statements often appear in a form other than that of the quoted references.     

A class of 4-pseudomanifolds similar to the lens spaces

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these lens-like spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out     

A class of 4-pseudomanifolds similar to the lens spaces

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these lens-like spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out     

Cellularity in polyhedra

A compact subset X of a polyhedron P is cellular in P if there is a pseudoisotropy of P shrinking precisely X to a point. A proper surjection between polyhedra f:P→Q is cellular if each point inverse of f is cellular in P. It is shown that if f:P→Q is a cellular map and either P or Q is a generalized n-manifold, n≠4, then f is approximable by homeomorphisms. Also, if P or Q is an n-manifold with boundary, n≠4, 5, then a cellular map f:P→Q is approximable by homeomorphisms. A cellularity criterion for a special class of cell-like sets in polyhedra is established.     

Characterizing Hilbert cube manifolds by their homological structure

A hierarchy of disjoint ?ech carriers properties is introduced; and each is shown to be characteristic of ANR's whose products with 2-cells are Hilbert cube manifolds.     

An approximation theorem in shape theory

In this paper it is shown that if X is a compactum in the interior of a PL manifold M and if U is a neighborhood of X in M, then there is a compactum X′ in U such that X and X′ have the same relative shape in U and the embedding dimension of X′ equals the fundamental dimension of X. Whenever the dimension of M is not equal to three, the relative shape equivalence from X′ to X can be realized by an infinite isotopy of M.     

Extensions with positive real part. A new version of the abstract band method with applications

This paper presents a new version of the abstract band method. The new scheme applies to extension problems for classes of essentially bounded functions, continuous functions, and bounded operators, which were not covered by earlier versions of the abstract band method.     

Note on a Theorem of Munkres

We prove that given a Riemannian manifold with boundary, having a finite number of compact boundary components, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result extends to manifolds and to embedded PL manifolds of dimensions 2, 3 and 4. We employ these results to prove that manifolds of the types above admit quasimeromorphic mappings onto As an application we prove the existence of G-automorphic quasimeromorphic mappings, where G is a Kleinian group acting on Dedicated to the memory of Robert BrooksThis paper represents part of the authors Ph.D. thesis written under the supervision of Prof. Uri Srebro.     

Folding a surface to a polygon

A concept of folding for compact connected surfaces, involving the partition of the surface into combinatorially identical n-sided topological polygons, is defined. The existence of such foldings for given n and given surfaces is explored, with definitive results for the sphere and the torus. We obtain necessary conditions for the existence of such foldings in all other cases.Supported by Kuwait University Grant SM 043.